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318 /EOLIAN TONES [394
We may now compare this with the known values of v for the temperatures in question. We have
/*m = 1853 x 10~7, Psa = -001161,
(j.u = 1765 x 10~7, Pll = -001243; so that i,2 = -1596, ,^=-1420,
and v2 — i>! = '018.
The difference in the values of v at the two temperatures thus accounts in (6) for the change of frequency both in sign and in order of magnitude.
As regards dynamical explanation it was evident all along that the origin of vibration was connected with the instability of the vortex sheets which tend to form on the two sides of the obstacle, and that, at any rate when a wire is maintained in transverse vibration, the phenomenon must be unsym-metrical. The alternate formation in water of detached vortices on the two sides is clearly described by H. Benard*. "Pour une vitesse suffisante, au-dessous de laquelle il "n'y a pas de tourbillons (cette vitesse limite croit avec la viscosite et decroit quand 1'epaisseur transversale des obstacles aug-mente), les tourbillons products periodiqitement se detachent alternativement a droite et & gauche du remous d'arriere qui suit le solide; Us gagnent presque immediatement' leur emplacement definitif. de sorte qu'd Varriere de Vobstacle se forme une double rangde alternSe d'entonnoirs stationnaires, ceux de droite dextrogyres, ceux de gauche levogyres, separe's par des intervalles dgauoo"
The symmetrical and- unsymmetrical processions of vortices were also figured by Mallockf from direct observation.
In a remarkable theoretical investigation;}; Karman has examined the question of the stability of such processions. The fluid is supposed to be incompressible, to be devoid of viscosity, and to move in two dimensions. The vortices are concentrated in points and are disposed at equal intervals (I) along two parallel lines distant h. Numerically the vortices are all equal, but those on different lines have opppsite signs.
Apart from stability, steady motion is possible in two arrangements (a) and (6), fig. 1, of which (a) is symmetrical. Karman shows that (a) is always unstable, whatever may be the ratio of h to I; and further that (6) is usually ( unstable also. The single exception occurs when cosh (rrh/l) = ^2, or h/l = 0'283. With this ratio of hjl, (b) is stable for every kind of displacement except one, for which there is neutrality. The only procession which can possess a practical permanence is thus defined.
* G. R. t. 147, p. 839 (1908). f Proc. Roy. Soc. Vol. LXXXIV. A, p. 490 (1910).
J GGttingen Nachrichten, 1912, Heft 5, S. 547; Karman aud Rubach, PhysiJi. Zeitschrift, 1912, p. 49. I have verified the more important results.=/ca0 . F (v/«a2, ca3)."1892); Scientific Papm, Vol. iv. j>. Ifl. t Ann. der Plnjsik, Vol. xx. p. 848 (1906).